Position Sensorless Motor Control

ABSTRACT

A control system is provided for an AC electric motor which comprises a rotor and a stator and a plurality of phase windings connected in a star formation, each winding having one end connected to a common neutral point and another end arranged to have a terminal voltage applied to it. The control system comprises switching means arranged to control the terminal voltages applied to the windings and control means arranged to control the switching means so as to switch it between a plurality of states in each of a sequence of PWM periods. The control means is further arranged to measure the voltage at the neutral point at sample times within the PWM periods and to generate from the measured voltages an estimation of the rotational position of the rotor.

INTRODUCTION

The present invention relates to the control of electrical machines. Theinvention is primarily targeted at salient permanent magnet sinusoidalmotors (PMSMs) with low per-unit mutual inductance. However, it isapplicable to any type of sinusoidal machine (including induction motorsand synchronous reluctance motors) that have self and mutual inductancesthat vary with rotor position.

BACKGROUND TO THE INVENTION

Various methods are known for determining the rotational position of anelectrical machine without using a dedicated position sensor. These areknown as sensorless techniques. Many known methods use current sensorsarranged to measure the current in the coils of the machine togetherwith algorithms that can determine the position of the motor from thecurrent measurements. Examples of such systems are described, forexample in WO2004/023639A and WO2006/037966.

SUMMARY OF THE INVENTION

The present invention provides a control system for an AC electricmachine, such as a motor, which comprises a rotor and a stator and aplurality of phase windings connected in a star formation, each windinghaving one end connected to a common neutral point and another endarranged to have a terminal voltage applied to it. The control systemcomprises switching means which may be arranged to control the terminalvoltages applied to the windings, and control means, which may bearranged to control the switching means, for example so as to switch itbetween a plurality of states in each of a sequence of PWM periods. Thecontrol means may be further arranged to measure the voltage at theneutral point at sample times, which may for example be within the PWMperiods, and to generate from the measured voltages an estimation of therotational position of the rotor.

The neutral point voltage may include a component that varies with theinductance of the windings, and therefore with the position of therotor. It may also include other components, such as a DC component andharmonics. In some cases, for example where this component that varieswith position is small, the control means may be arranged to perform oneor more steps to extract this component from the measured neutral pointvoltage and use this extracted component of the measured voltage toestimate the motor position.

The control means may be further arranged to measure a DC link voltageand to further use the DC link measurement to generate the positionestimation.

The control means may be arranged to measure the neutral point voltagewhen all of the phases are connected to the DC link voltage thereby byto generate an estimation of the DC link voltage, and to use thatestimation to generate the position estimation.

The control means may be arranged to compare the measured DC linkvoltage with the estimated DC link voltage and if they fail to meet aconsistency condition, to generate a fault indication.

The sample times may be within the PWM periods. However in some casesthey may be outside the normal torque-generating PWM periods duringtimes when voltages are applied to the windings specifically for thepurpose of position estimation.

The control means may be arranged to measure the neutral point voltageduring two complementary states within one PWM period, during two pairsof two complementary states within one PWM period, or during each of anumber of non-complementary states equal to the number of phases of themotor, within one PWM period. Where the machine is a three-phase machinethis will be three active states.

The control means may be arranged to define a minimum state time forneutral point voltage measurement and a minimum number of active stateswithin a PWM period in which the neutral point voltage needs to bemeasured. It may be arranged to determine a demanded net voltage for thePWM period, and to control the switching means so as to provide at leastthe minimum number of active states, each for at least the minimum statetime. It may also be arranged to control the switching pattern so as toprovide the demanded net voltage.

The present invention further provides a method of sensing the positionof a sinusoidal machine comprising measuring a neutral-point voltage inat least two inverter states and determining motor position from themeasured voltages.

The present invention is best suited to motors with low per-unit mutualinductance, but may also be suitable for motors with higher mutualinductance. It is therefore particularly well suited for motors withconcentrated windings as these tend to have lower mutual inductance thandistributed windings. It is equally applicable to systems with a singlecurrent sensor in the DC link and with multiple phase current sensors.It is suitable for any drive application but is particularly attractivefor sensorless electric drive applications where low acoustic noise isrequired. Some embodiments are in particular suitable for use with PWMinjection sensorless control.

The original target application for the present invention is a salientPMSM with concentrated windings and low per-unit mutual inductance witha single current-sensor drive for an automotive application.

Some embodiments of the present invention differ from other techniquesthat attempt to derive the electrical position from the neutral pointvoltage in that the resolution is far greater, resolving position downto levels associated with carrier frequency injection techniques, and isonly limited by the resolution of the measurement hardware.

An example of an existing technique which uses the neutral point voltageis the ELMOS VirtuHall product which produces an electrical positionsignal with a resolution of 30° (electrical).

The present invention further provides a control system for an ACelectric motor which comprises a rotor and a stator and a plurality ofphase windings connected in a star formation, each winding having oneend connected to a common neutral point and another end arranged to havea terminal voltage applied to it, the control system comprisingswitching means arranged to control the terminal voltages applied to thewindings and conirol means arranged to control the switching means so asto switch it between a plurality of states in each of a sequence of PWMperiods, wherein the control means is further arranged to measure thevoltage at the neutral point when all of the phases are connected to theDC link voltage thereby by to generate an estimation of the DC linkvoltage.

The control means may be further arranged to compare a measured DC linkvoltage with the estimated DC link voltage and if they fail to meet aconsistency condition, to generate a fault indication.

Preferred embodiments of the present invention will now be described byway of example only with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram of a salient PM electric motor;

FIG. 2 is a schematic diagram of a three phase electrical model of themotor of FIG. 1 and associated control system;

FIG. 3 is a diagram of a typical PWM pattern in the system of FIG. 2that enables the neutral point voltage to be measured in four non-zeroinverter states;

FIG. 4 is a space vector diagram for the PWM pattern shown in FIG. 3;

FIG. 5 is a space vector diagram for a PWM pattern containing twoinverter states suitable for neutral-point sampling for low-speedsensorless control according to an embodiment of the invention;

FIG. 6 is a diagram of a PWM pattern containing only one inverter statesuitable for measuring neutral point voltage with centre-aligned PWM;

FIG. 7 is a space vector diagram showing where the conditions shown inthe example of FIG. 6 apply.

FIG. 8 is a space vector diagram for a further PWM pattern containinginverter states suitable for neutral-point sampling according to anotherembodiment of the invention;

FIG. 9 is a functional diagram of a control system according to anembodiment of the invention arranged to use the PWM pattern of FIG. 8;

FIG. 10 is a graph showing how the neutral point voltage in the systemof FIG. 2 varies with PWM state and motor position.

Referring to FIG. 1, a three phase electrically commutated sinusoidal ACbrushless permanent magnet synchronous motor comprises a rotor 2 having,for example, six magnets 4 mounted on it, which in this instance arearranged so as to provide six poles which alternate between north andsouth around the rotor 2. The rotor 2 therefore defines three direct ord axes evenly spaced around the rotor and three quadrature or q axesinterspaced between the d axes. A stator 106 in this embodimentcomprises a nine slot copper wound element having three groups 8A, 8B,8C of three teeth, each group of teeth having a common winding forming arespective phase. There are therefore three electrical cycles in eachfull rotation of the rotor, and the three teeth in any phase 8A, 8B, 8 care always in the same electrical position as each other. The number ofpoles and slots of the machine will vary depending on the application,and can affect its suitability for use with the present invention aswill be described in more detail below.

Referring to FIG. 2 the three stator windings 10, 12, 14 are associatedwith the three phases U, V and W of the motor. The windings 10, 12, 14are connected together in a star formation, each having one endconnected to a common neutral point 16 and a terminal 17 at the otherend connected to a drive circuit 18. Each winding has an inductanceL_(u), L_(v), L_(w) associated with it, a resistance R associated withit, a current i_(u), i_(v), i_(w) flowing through it, and a back EMFe_(u), e_(v), e_(w) generated in it, at any one time. The terminalvoltage applied to each of the windings by the drive circuit 18 at anytime is referred to as u_(u) u_(v) u_(w) respectively. The drive circuit18 is arranged to connect each of the windings 10, 12, 14 independentlyto either one of two voltage inputs, in this case either a positive DCvoltage +V or ground or 0V. To achieve this each winding 10, 12, 14 isconnected via a top transistor 20 to a top rail 21 which is connected tothe positive DC input +V from a DC power supply, and via a bottomtransistor 22 to a bottom rail 23 which is connected to the negativeterminal of the DC supply, which is at ground. The transistors 20, 22are controlled by a controller 24 which is arranged to control thetransistors 20, 22 to switch the motor between six active states and twoinactive states. These states are referred to as inverter states as thedrive circuit is often referred to as an inverter. Conventionally theactive states are numbered 1 to 6, one of the inactive states when allwindings 10, 12, 14 are connected to earth is referred to as 0 and theother inactive state when all of the windings are connected to +V isreferred to as 7. In this application each of the active states isreferred to by the letter of the winding which is connected to adifferent voltage to the other two, and by the sign of the voltage towhich that winding is connected. So for example the state where phase Uis connected to +V and V and W to 0V is referred to as +U, and the statewhere phase V is connected to 0V and phases U and W are connected to +Vis referred to as state −V. A full list of the active states is providedin the table below.

U terminal V terminal W terminal State name voltage voltage voltage +U+V 0 V 0 V −U 0 V +V +V +V 0 V +V 0 V −V +V 0 V +V +W 0 V 0 V +V −W +V+V 0 V

A voltage sensor 26 is arranged to measure the voltage at the neutralpoint 16 and provide a voltage signal as an input to the controller 24which varies with that neutral point voltage. Also voltage sensor 28 isconnected in a DC link between the two rails 21, 23 to measure thevoltage of the top rail 21 relative to the bottom rail 23, and hence thetotal voltage applied across the drive circuit 18. The controller 24also receives an input indicative of the drive that is required from themotor, for example in the form of a current demand, or a torque demand.The controller 24 is arranged to control the drive circuit using pulsewidth modulated PWM control in which, for each of a series of PWMperiods of equal length, the net voltage required is determined, andthen the combination of states and their duration required to producethe required net voltage are calculated and applied to the windings.

In a star-point connected motor such as that of FIG. 1, the three phasecurrents must sum to zero:

i _(u) +i _(v) +i _(w)=0  (1)

Furthermore, for a motor with perfect sinusoidal three-phase rotorback-EMFs, the sum of the rotor back-EMFs is also zero:

e _(u) +e _(v) +e _(w)=0  (2)

From FIG. 1 it can be seen that the voltage drop from the terminalvoltage of each phase (u_(u), u_(v), u_(w)) to the neutral (star) pointvoltage (u_(n)) is:

$\begin{matrix}{\mspace{79mu} {{{{{u_{u} - {L_{uu}\frac{i_{u}}{t}} - {M_{uv}\frac{i_{v}}{t}} - {M_{uw}\frac{i_{w}}{t}} - {i_{u}R} - e_{u}} = u_{n}}\mspace{79mu} {u_{v} - {M_{vu}\frac{i_{u}}{t}} - {L_{vv}\frac{i_{v}}{t}} - {M_{vw}\frac{i_{w}}{t}} - {i_{v}R} - e_{v}}} = u_{n}}{{u_{w} - {M_{wu}\frac{i_{u}}{t}} - {M_{wv}\frac{i_{v}}{t}} - {L_{ww}\frac{i_{w}}{t}} - {i_{w}R} - e_{w}} = u_{n}}}} & {\left( {3a} \right),\left( {3b} \right),\left( {3c} \right)}\end{matrix}$

Where L represents self-inductance and M represents mutual inductancesuch that L_(uu), L_(vv) and L_(ww) are the self inductances of phasesu, v and w respectively; M_(uv) is the mutual inductance between phase uphase v and so on. (Note that the coupling from phase u to v is the sameas from phase v to u, i.e. M_(uv)=M_(vu))

Summing equations (3a), (3b) and (3c) and substituting equations (1) and(2) to eliminate the ohmic and rotor back-EMF voltage drops, the neutralpoint voltage is then:

$\begin{matrix}{u_{n} = {\frac{1}{3}\begin{bmatrix}{u_{u} + u_{v} + u_{w} - {\left( {L_{uu} + M_{uv} + M_{uw}} \right)\frac{i_{u}}{t}} -} \\{{\left( {M_{uv} + L_{vv} + M_{vw}} \right)\frac{i_{v}}{t}} - {\left( {M_{uw} + M_{vw} + L_{ww}} \right)\frac{i_{w}}{t}}}\end{bmatrix}}} & (4)\end{matrix}$

In general for an ideal non-salient motor with three-phase sinusoidallydistributed windings the mutual inductance is minus half theself-inductance, that is:

$\begin{matrix}{M = {{- \frac{1}{2}}L}} & (5)\end{matrix}$

In which case the mutual and self-inductances in equation (4) sum up tozero and the neutral point voltage simplifies to:

$\begin{matrix}{u_{n} = {\frac{1}{3}\left\lbrack {u_{u} + u_{v} + u_{w}} \right\rbrack}} & (6)\end{matrix}$

In reality mutual inductance is always slightly less than 50% of theself inductance due to practical effects such as flux leakage, althoughfor most conditions the ideal conditions stated above can be assumed.

However, for motors with concentrated windings it is possible to achievemutual inductances significantly lower than 50% of that self inductance.For example, for some motors it is possible to design a salient PMSMwith a mutual inductance that is less than 10% of the self-inductance,such as a 12 slot 10 pole motor. In which case the mutual inductanceterms can be eliminated and equation (4) approximates to:

$\begin{matrix}{u_{n} \approx {\frac{1}{3}\left\lbrack {u_{u} + u_{v} + u_{w} - {L_{uu}\frac{i_{u}}{t}} - {L_{vv}\frac{i_{v}}{t}} - {L_{ww}\frac{i_{w}}{t}}} \right\rbrack}} & (7)\end{matrix}$

Hence with low mutual inductance motors extra terms appear in theneutral point voltage that depend on the incremental inductance of themotor phases. Assuming an unsaturated salient motor with sinusoidalinductance distribution, these inductances will be of the form:

$\begin{matrix}{\begin{pmatrix}L_{uu} \\L_{vv} \\L_{ww}\end{pmatrix} = \begin{pmatrix}{\overset{\_}{L} - {\overset{}{L}{\cos \left( {2\theta_{e}} \right)}}} \\{\overset{\_}{L} - {\overset{}{L}{\cos \left( {{2\theta_{e}} - \frac{4\pi}{3}} \right)}}} \\{\overset{\_}{L} - {\overset{}{L}{\cos \left( {{2\theta_{e}} - \frac{8\pi}{3}} \right)}}}\end{pmatrix}} & (8)\end{matrix}$

Where for a PMSM

$\begin{matrix}{{\overset{\_}{L} = {\frac{1}{2}\left( {L_{q} + L_{d}} \right)}}{\overset{}{L} = {\frac{1}{2}\left( {L_{q} - L_{d}} \right)}}} & {\left( {9a} \right).\left( {9b} \right)}\end{matrix}$

Where L_(d) is the direct-axis inductance and L_(q) is thequadrature-axis inductance. In a salient PMSM L_(q)>L_(d).

Combining equations (7) and (8) and eliminating the mean component ofthe inductances using equation (1), it can be seen that the neutralpoint voltage contains position-dependent information that could be usedfor sensorless control:

$\begin{matrix}{u_{n} = {\frac{1}{3}\begin{bmatrix}{u_{u} + u_{v} + u_{w} -} \\{\overset{}{L}\begin{pmatrix}{{{\cos \left( {2\theta_{e}} \right)}\frac{i_{u}}{t}} +} \\\begin{matrix}{{{\cos \left( {{2\theta_{e}} - \frac{4\pi}{3}} \right)}\frac{i_{v}}{t}} +} \\{{\cos \left( {{2\; \theta_{e}} - \frac{8\pi}{3}} \right)}\frac{i_{w}}{t}}\end{matrix}\end{pmatrix}}\end{bmatrix}}} & (10)\end{matrix}$

This position dependent component of the neutral point voltage will berelatively large for motors with low mutual inductance. For motors withhigh mutual inductance it is still present, but is relatively small, andmay need to be extracted from the total signal and amplified if it is tobe used for position sensing.

Techniques for Deriving Motor Position from Neutral-Point Voltage

The neutral point voltage can be sampled in specific inverter states toobtain Motor position information from a salient PMSM with low mutualinductance.

The neutral point voltage in equation (10) contains position-dependentcomponents that are proportional to the rate of change of current in themotor windings. The rate of change of current for each winding will inturn depend on the inverter state.

Full closed-form analysis of this effect is not straightforward becauseof the complexity of the interactions caused by the position-dependentnature of the inductances. However, a good understanding of thebehaviour can be gained by considering the rate of change of current ina non-salient motor with inductance equal to the mean phase inductance,L. This gives an approximation to the behaviour of the rates of changeof current that is sufficient to gain an understanding of the sensorlesstechnique. Numerical simulation that does not make use of thesesimplifying assumptions shows that this approximation is sufficient.

These rates of change of current in a non-salient machine are given by:

$\begin{matrix}{{\frac{}{t}\begin{pmatrix}i_{u} \\i_{v} \\i_{w}\end{pmatrix}} \approx {{\begin{pmatrix}{{- R}/\overset{\_}{L}} & 0 & 0 \\0 & {{- R}/\overset{\_}{L}} & 0 \\0 & 0 & {{- R}/\overset{\_}{L}}\end{pmatrix}\begin{pmatrix}i_{u} \\i_{v} \\i_{w}\end{pmatrix}} + {\begin{pmatrix}{{- 1}/\overset{\_}{L}} & 0 & 0 \\0 & {{- 1}/\overset{\_}{L}} & 0 \\0 & 0 & {{- 1}/\overset{\_}{L}}\end{pmatrix}\begin{pmatrix}e_{u} \\e_{v} \\e_{w}\end{pmatrix}} + {\frac{1}{3\overset{\_}{L}}\begin{pmatrix}2 & {- 1} & {- 1} \\{- 1} & 2 & {- 1} \\{- 1} & {- 1} & 2\end{pmatrix}\begin{pmatrix}u_{u} \\u_{v} \\u_{w}\end{pmatrix}}}} & (11)\end{matrix}$

During inverter state +U, where phase U is high and phase V and W arelow, the terminal voltages of the motor are u_(u)=u_(DC) andu_(v)=u_(w)=0, where u_(DC) is the DC link voltage. The rate of changeof current for each phase in inverter state +U is approximately:

$\begin{matrix}{{\frac{i_{u}}{t}_{+ U}{\approx {{\frac{- R}{\overset{\_}{L}}i_{u}} - {\frac{1}{\overset{\_}{L}}e_{u}} + {\frac{2}{3\overset{\_}{L}}U_{DC}}}}}{\frac{i_{v}}{t}_{+ U}{\approx {{\frac{- R}{\overset{\_}{L}}i_{v}} - {\frac{1}{\overset{\_}{L}}e_{v}} - {\frac{1}{3\overset{\_}{L}}U_{DC}}}}}{\frac{i_{w}}{t}_{+ U}{\approx {{\frac{- R}{\overset{\_}{L}}i_{w}} - {\frac{1}{\overset{\_}{L}}e_{w}} - {\frac{1}{3\overset{\_}{L}}U_{DC}}}}}} & {\left( {12a} \right)\text{-}\left( {12c} \right)}\end{matrix}$

Likewise, the rates of change of current during the −U inverter stateare:

$\begin{matrix}{{\frac{i_{u}}{t}_{- U}{\approx {{\frac{- R}{\overset{\_}{L}}i_{u}} - {\frac{1}{\overset{\_}{L}}e_{u}} - {\frac{2}{3\overset{\_}{L}}U_{DC}}}}}{\frac{i_{v}}{t}_{- U}{\approx {{\frac{- R}{\overset{\_}{L}}i_{v}} - {\frac{1}{\overset{\_}{L}}e_{v}} + {\frac{1}{3\overset{\_}{L}}U_{DC}}}}}{\frac{i_{w}}{t}_{- U}{\approx {{\frac{- R}{\overset{\_}{L}}i_{w}} - {\frac{1}{\overset{\_}{L}}e_{w}} + {\frac{1}{3\overset{\_}{L}}U_{DC}}}}}} & {\left( {13a} \right)\text{-}\left( {13c} \right)}\end{matrix}$

It can be seen that these terms are dependent on speed (rotor back-EMFmagnitude) and load (phase current magnitude). For optimum sensorlessposition estimation these speed and load terms need to be removed. Thiscan be achieved either by applying model-based corrections in real-timeor by eliminating them by combining more than one measurement. For easeof analysis the zero speed/zero current condition can be consideredbecause this removes the terms based on current i and EMF e from theanalysis. For the specific inverter state +U equations 12a to 12c can besubstituted into equation (10) and the neutral point voltage can beshown to be:

$\begin{matrix}{{u_{n}_{+ U}} = {\frac{U_{DC}}{3}\left\lbrack {1 - {\frac{\overset{}{L}}{\overset{\_}{L}}{\cos \left( {2\theta_{e}} \right)}}} \right\rbrack}} & (14)\end{matrix}$

Similarly for the −U state the neutral point voltage can be approximatedby substituting equations 13a to 13c into equation 10 to give:

$\begin{matrix}{{u_{n}_{- U}} = {\frac{U_{DC}}{3}\left\lbrack {2 + {\frac{\overset{\_}{L}}{\overset{\_}{L}}{\cos \left( {2\theta_{e}} \right)}}} \right\rbrack}} & (15)\end{matrix}$

This approach can be extended to the remaining phases V and W.

For conditions where the motor speed and current are not zero theelectrical position information is still present, it is just thatadditional terms are present which introduce additional harmonics. Theseadditional harmonics can be removed by the use of feedforwardcompensation terms, a technique that is common practice and well knownto the skilled man.

For motors where the mutual inductance is approaching 50%, for example a9 slot 6 pole motor, it is possible to boost the differential part ofthe neutral point voltage which gives the position information, byremoving the DC component of the signal. Even though the variation dueto position is very small the amplification and subsequent filtering ofthe position-dependent component of the neutral point voltage signal isenough to obtain the electrical position.

PWM Modulation Strategies that Facilitate Sensorless Position Estimation

Equations 14 and 15 show how rotor position information may be obtainedby sampling the neutral-point voltage during different inverter states.

One way to create the required inverter states would be to interrupt thePWM sequence and inject known inverter states for one or more PWMperiods in order to allow the required measurements to be taken. This isequivalent to the basic INFORM technique, except that the INFORMtechnique measures rate of change of current rather than neutral pointvoltage (and optionally the DC link voltage) during the applied inverterstates. The major disadvantage of the INFORM-style approach is thatinterrupting the PWM pattern for one or more PWM periods usually resultsin sub-harmonics of the switching pattern that lead to considerablelevels of acoustic noise.

A preferable approach is to measure the neutral point voltage within theinverter states within a normal PWM period. This can be achieved withoutreducing the effective switching frequency of the transistors, and sotherefore results in little or no extra acoustic noise which is a majorbenefit for numerous applications. This makes it possible to achievesilent low-speed sensorless control. There are many different PWMmodulation strategies that would facilitate such an approach. Someexamples of these are described below. This technique is similar in thisrespect to the low-acoustic noise low-speed sensorless control describedin WO2004/023639A, except that the technique described in that documentmeasures the rate of change of phase current (di/dt) during particularinverter states, whereas some embodiments of the present invention takemeasurements of the neutral-point voltage (and optionally the DC linkvoltage) during particular inverter states.

Sampling U_(n) in Four Inverter States The electrical motor positionθ_(e) may be fully determined by combining measurement of the DC linkvoltage (UDC) with measurements of the neutral point voltage (u_(n))taken during two pairs of complementary non-zero inverter states. Heretwo states are referred to as complementary when the phase voltages arereversed between them, such as +U and −U. Using equations 14 and 15 itcan be shown that:

$\begin{matrix}{{{{u_{n}\left( {+ U} \right)} - {u_{n}\left( {- U} \right)}} \approx {\frac{U_{DC}}{3}\left\lbrack {{- 1} - {\frac{2\overset{}{L}}{\overset{\_}{L}}{\cos \left( {2\theta_{e}} \right)}}} \right\rbrack}}{{{u_{n}\left( {+ W} \right)} - {u_{n}\left( {- W} \right)}} \approx {\frac{U_{DC}}{3}\left\lbrack {{- 1} - {\frac{2\overset{}{L}}{\overset{\_}{L}}{\cos \left( {{2\theta_{e}} - \frac{8\pi}{3}} \right)}}} \right\rbrack}}} & {\left( {16a} \right),\left( {16c} \right)}\end{matrix}$

There are two equations and seven independent variables. By measuringu_(n)(+U), u_(n)(−U), u_(n)(+W), u_(n)(−W) and U_(DC) five of thevariables are known and the two equations may be rearranged to yield2θ_(e).

A convenient method for making this calculation is as follows:

$\begin{matrix}{{u_{nU} = {{u_{n}\left( {+ U} \right)} - {u_{n}\left( {- U} \right)} + \frac{U_{DC}}{3}}}{u_{nW} = {{u_{n}\left( {+ W} \right)} - {u_{n}\left( {- W} \right)} + \frac{U_{DC}}{3}}}{u_{nV} = {{- u_{nU}} - u_{nW}}}{{2\hat{\theta}} = {{{- \arg}\left\{ {u_{nU} + {u_{nV}^{j\frac{2\pi}{3}}} + {u_{nW}^{j\frac{4\pi}{3}}}} \right\}} + \pi}}} & {\left( {17a} \right)\text{-}\left( {17d} \right)}\end{matrix}$

Where {circumflex over (θ)} is the position estimate and ‘arg’ is theargument or angle of a complex number.

Conventional centre-aligned PWM techniques produce a maximum of twonon-zero inverter states within a single PWM period, and are thereforenot suitable for use with this technique. However, the required inverterstates can be achieved by judicious shifting of the PWM patterns asshown in FIG. 3. It will be appreciated that a certain minimum time isrequired at each inverter state transition to allow time for the neutralpoint voltage measurement to settle and then to be sampled. This time isdesignated T_(min) and depends on transient factors such as straycapacitance, electromagnetic pickup on signals, signal conditioning etc.

PWM Pattern Realisation for Four Inverter States

FIG. 3 shows an example of a PWM pattern that produces four activestates in one PWM period. The terminal voltages of the three phases U, Vand W over the period are shown, with the resulting states indicated foreach part of the PWM period, and the times at which the neutral pointvoltage is sampled are also shown. In this example the total modulationindex demand, i.e. the net voltage during the PWM period, is zero (whichin this case is achieved by commanding a 50% duty cycle in each phase)but by shifting the PWM patterns from their original centre-alignedconfiguration the four required inverter states are produced atdifferent points within the PWM period. Note that in this case the PWMedges are biased around the centre of the PWM waveform, but they couldjust as easily be biased to the start of the waveform (e.g. with phase Ugoing high at the start of the PWM period) without changing theunderlying principle.

Note, the states which are normally represented as numbers 1-6 arerepresented instead with letters and signs described above that showwhich phase they affect.

As with most low-speed sensorless techniques, the shifting of thepatterns results in a reduction in the maximum realisable voltage. Thisis demonstrated in terms of space vectors in FIG. 4, in which the samenomenclature is used to define the different states.

The resulting maximum modulation index (defined in terms of the ratio ofpeak phase voltage to half the DC link voltage) for the case where theminimum time required in each inverter state to measure the neutralpoint voltage is T_(min), is

$\begin{matrix}{M_{\max} = {\frac{2}{\sqrt{3}}\left( {1 - \frac{4T_{\min}}{T_{PWM}}} \right)}} & (18)\end{matrix}$

where TPWM is the PWM period. For a typical example where TPWM=50 μs andT_(min)=3 μs, this results in a maximum modulation index of 1.01. Sincethis technique is only used at low speeds where full modulation index isnot required, this should be acceptable.

One advantage of the neutral-voltage sensing technique over di/dtsensing techniques such as that descried in WO 2004/023639 A1 is thatless time is required for the signal to settle, which means that T_(min)will be smaller and a higher maximum modulation index may be achieved.

In certain applications where single current sensing is used it may benecessary to have a larger minimum inverter state time for two of theinverter states (e.g. +V & −W) to allow time for current measurements tobe taken for standard current control. In this case equation (19) wouldhave to be modified accordingly.

DC Link Voltage Measurement

The above embodiment requires a measurement of the DC link voltage aspart of the position calculation. This can be determined in a number ofways. One method is to filter the DC link voltage in the analogue domainwith a filter time-constant that is significantly (e.g. many times)longer than the PWM period. This enables the link voltage to be sampledat any point in the PWM period. Alternatively an unfiltered DC linkvoltage (or a filtered version of the link voltage with a very lowfilter time constant) can be sampled during each if the inverter states.The latter technique may be necessary in systems where the DC linkresistance is relatively high, causing the link voltage at the samplepoint to vary with phase current.

Both of these approaches could potentially suffer from the fact that twodifferent sensors are used to calculate position (neutral-point voltageand DC link voltage). If there are difficulties matching the performanceof these two sensors (e.g. gain and offset) then errors could result. Analternative way to determine DC link voltage that can overcome thisissue is to sample the neutral point voltage during the inverter state 7(see FIG. 3) where all three phases are high and the neutral pointvoltage is equal to the DC link voltage. Thus the same sensor is usedfor all five measurements. This approach can also be potentiallybeneficial in high-voltage systems where saving one voltage transducercould reduce the cost of the system.

The advantages of measuring the neutral point using two pairs ofinverter state measurements are that the first order harmonic can becancelled out. The disadvantage is that the additional measurementstates can limit the maximum voltage that can be applied.

Another disadvantage is that to compensate for the fact that nomeasurements are made for one of the three phases, the mean inductanceat the point of measurement should be known. This can only be measuredat the point where the two phases being measured change—at that pointall three phases have just been measured. As the mean inductance is onlyupdated periodically any change in the mean between updates willintroduce an error into the position measurement.

Ideally measurement of the DC link voltage should remove this error byremoving the DC link voltage component to leave just the harmoniccomponent but due to measurement errors this may not be the case.

Sampling u_(n) in Two Inverter States

Although ideally the neutral point voltage should be measured in twocomplementary pairs of inverter states, there are situations where itwould be desirable to derive position from measurements carried outduring just two non-complementary inverter states (for example state +Uand −W) as shown in FIG. 5. One reason for doing this is to increase themaximum modulation index; another reason is to minimise or eliminate thechange in to an existing modulation strategy such as centre-aligned PWM.Examples of these are discussed below.

When only two non-complementary inverter states are used, the ohmic androtor back-EMF voltage contributions are no longer cancelled out. Forexample, the neutral point voltage in a low mutual-inductance PMSMduring inverter state +U becomes (assuming the d axis current, i_(d), iszero):

$\begin{matrix}{{u_{n}\left( {+ U} \right)} \approx {{\frac{U_{DC}}{3}\left\lbrack {1 - {\frac{\overset{}{L}}{\overset{\_}{L}}{\cos \left( {2\theta} \right)}}} \right\rbrack} - {{\frac{\overset{\_}{L}}{\overset{\_}{L}}\left\lbrack {{i_{q}R} - \frac{k_{e}\omega_{m}}{\sqrt{3}}} \right\rbrack}{\cos \left( {{3\theta} + \frac{\pi}{2}} \right)}}}} & (19)\end{matrix}$

Where i_(q) is the q axis current, k_(e) is the rotor back-EMF constantand ω_(m), is the rotor mechanical frequency.

The ohmic and back-EMF voltage drops introduce voltage distortion on theneutral-point voltage signal at a frequency three times the electricalfrequency of the rotor. The extent of this distortion will depend on themachine design and operating conditions. For some applications(particularly high voltage applications where the effect of phaseresistance can be neglected) the distortion effect will be small and maybe ignored. In these circumstances the two neutral point voltagesmeasurements approximate to:

$\begin{matrix}{{{u_{n}\left( {+ U} \right)} \approx {\frac{U_{DC}}{3}\left\lbrack {1 - {\frac{\overset{}{L}}{\overset{\_}{L}}{\cos \left( {2\theta_{e}} \right)}}} \right\rbrack}}{{u_{n}\left( {- W} \right)} \approx {\frac{U_{DC}}{3}\left\lbrack {2 + {\frac{\overset{}{L}}{\overset{\_}{L}}{\cos \left( {{2\theta_{e}} - \frac{8\pi}{3}} \right)}}} \right\rbrack}}} & {\left( {20a} \right),\left( {20b} \right)}\end{matrix}$

And the position estimate, {circumflex over (θ)}, may be calculatedfrom:

$\begin{matrix}{{u_{nU} = {{u_{n}\left( {+ U} \right)} - \frac{U_{DC}}{3}}}{u_{nW} = {{- {u_{n}\left( {- W} \right)}} + \frac{2U_{DC}}{3}}}{u_{nV} = {{- u_{nU}} - u_{nW}}}{{2\hat{\theta}} = {{{- \arg}\left\{ {u_{nU} + {u_{nV}^{j\frac{2\pi}{3}}} + {u_{nW}^{j\frac{4\pi}{3}}}} \right\}} + \pi}}} & {\left( {21a} \right)\text{-}\left( {21d} \right)}\end{matrix}$

where ‘arg’ is the argument or angle of a complex number.

For other applications it may be necessary to compensate for the ohmicand back-EMF voltage drops. The ohmic voltage drop may be calculatedusing phase current measurements and knowledge of the phase resistance;the rotor back-EMF may be calculated using rotor speed and knowledge ofthe rotor back-EMF constant. These terms can then be used to compensateequation (19) to remove load- and speed-dependency. Alternativelyfeedforward compensation terms may be used to remove the unwantedharmonic components.

Note that the linear description of the ohmic and rotor back-EMF termsis a simplification and in reality a more complicated non-linear modelmay be required that accounts for effects such as inverter and magneticnon-linearities in order to achieve good compensation. Ideally, a singlemodel would be sufficient to cover all of the part-to-part variationswithin a product. However, if necessary the model could be calibrated oneach part at the end of the production line. Alternatively, aself-adaption algorithm could be used that learns and eliminates theohmic and back-EMF terms on-line whilst the drive is operating. Such analgorithm would take advantage of the fact that the ohmic and back-EMFvoltage terms in the neutral-point voltage vary at three times theelectrical frequency, and would use techniques such as synchronousrectification and filtering to isolate the undesired components andeliminate them.

PWM Pattern Realisation for Two Inverter States—Shifting PWMs

In order to guarantee that the two required inverter states are alwayspresent it is possible to shift the PWM signals in the same way asdescribed above. However, because only two inverter states are requireda higher maximum modulation index may be achieved.

Although any two inverter states from different phases could be used,the optimal solution is to use the two inverter states adjacent to theSVM sector where the voltage demand lies as shown in FIG. 5. In whichcase the maximum achievable modulation index demand becomes:

$\begin{matrix}{M_{\max} = {\frac{4}{3}\left( {1 - \frac{T_{\min}}{T_{PWM}}} \right)}} & (22)\end{matrix}$

Thus very high modulation indices are possible with this technique.

This PWM shifting technique is the same as is used to measure the statorcurrents in many single current sensor techniques. This makes thistechnique particularly well suited to single current sensor systems asno further modifications are required to the PWM algorithm. However, itshould be noted that this technique is equally applicable to systemswith multiple phase current sensors.

PWM Pattern Realisation for Two Inverter States—Centre- or Edge-AlignedPWMs

Although it is preferable to shift the PWM patterns as described above,it is possible to achieve two non-zero inverter states usingconventional PWM patterns (e.g. centre-aligned or edge-aligned) withoutany further modification since these patterns will contain the tworequired states under many conditions. However, an example is shown inFigure of a net voltage where it is not possible to produce two activestates of sufficient length. In the example shown the −W state is tooshort to enable the neutral-point voltage to be sampled. FIG. 7 showsthe parts of the vector space where, as with the net voltage required inFIG. 6, it is not possible to produce two inverter states of sufficientlength. The light shaded bands between the broken lines are the regionswhere there exists only one non-zero inverter state of sufficient lengthfor sampling; the dark shaded central star shaped area is the regionwhere there are no non-zero inverter state of sufficient length forsampling.

One of the ways that this problem is overcome in certain di/dtsensorless schemes, and which can also be used in embodiments of thisinvention, is to modify the PWM patterns over two consecutive PWMperiods. In one of those periods the duty cycle demands are modified sothat there is sufficient time to measure the neutral point voltage inboth inverter states; in the other period the duty cycle demands aremodified by an equal and opposite amount so that the total voltagedemand over the two PWM cycles is equal to the original demand. Theproblem with this technique is that it will introduce high-frequencyphase current components that are sub-harmonics of the PWM frequency,which is likely to lead to acoustic noise. However, if the time requiredto sample the neutral-point voltage, T_(min) is sufficiently small, themagnitude of the acoustic noise may be negligible. Again this is anadvantage of the neutral-point technique over di/dt techniques since itwill generally be the case that the time required for the neutral-pointmeasurement to settle will be less than the equivalent di/dtmeasurement.

An alternative solution for the regions where only one non-zero inverterstate of sufficient length can be realised is to track the positioninformation from just one inverter state. Although a unique positioncannot be determined from a single state, it is possible to estimate ortrack the position based on knowledge of the last known position wheretwo inverter states were measurable and the change in the voltage forthe one measurable state. However, this technique would not work at verylow modulation index demands where there are no non-zero inverter statesof sufficient length to measure the neutral point voltage (see Figure).In this situation an alternative technique would be required.

If edge-aligned modulation is used instead of centre-aligned modulation,the size of the operating region that cannot realise two inverter statesof sufficient length is halved. In this case the magnitude of theeffects described above is also halved which gives edge-alignedmodulation an advantage over centre-aligned modulation in this instance.

It will be appreciated that by reducing the number of measurements toonly two inverter states the maximum voltage can be realised isincreased, but without the removal of the resistance term additionalcompensation is required.

By measuring the inverter in two states the sensorless algorithm canoperate across the full speed/load range of the motor and therefore asingle sensorless algorithm can be used. Traditionally two algorithmsare required—a low speed technique (which is where this algorithm isintended to operate) and a high speed technique which uses the back EMFto determine electrical position. This embodiment can remove the needfor a separate high speed technique.

As with the embodiment using four inverter state measurement knowledgeof the correct mean inductance is important and any error in this willresult in position error.

Sampling u_(n) in Six Inverter States

It is possible to create PWM waveforms with six non-zero inverterstates.

With neutral-point voltage measurements in all six non-zero voltagestates the position is defined by equations (16a), (16b) and (16c):

$\begin{matrix}{\mspace{79mu} {{{{u_{n}\left( {+ U} \right)} - {u_{n}\left( {- U} \right)}} \approx {\frac{U_{DC}}{3}\left\lbrack {{- 1} - {\frac{2\overset{}{L}}{\overset{\_}{L}}{\cos \left( {2\theta_{e}} \right)}}} \right\rbrack}}{{{u_{n}\left( {+ V} \right)} - {u_{n}\left( {- V} \right)}} \approx {\frac{U_{DC}}{3}\left\lbrack {{- 1} - {\frac{2\overset{}{L}}{\overset{\_}{L}}{\cos \left( {{2\theta_{e}} - \frac{4\pi}{3}} \right)}}} \right\rbrack}}{{{u_{n}\left( {+ W} \right)} - {u_{n}\left( {- W} \right)}} \approx {\frac{U_{DC}}{3}\left\lbrack {{- 1} - {\frac{2\overset{}{L}}{\overset{\_}{L}}{\cos \left( {{2\theta_{e}} - \frac{8\pi}{3}} \right)}}} \right\rbrack}}}} & {\left( {16a} \right),\left( {16b} \right),\left( {16c} \right)}\end{matrix}$

This makes it possible to determine position solely from the neutralpoint voltage signals without needing to measure the DC link voltage.This means that only one voltage sensor is needed to determine motorposition. This could be important if the variation in performancebetween the DC link voltage sensor and the neutral-point voltage sensorare sufficiently high to cause problems with the position estimate. Italso reduces the required number and accuracy of the voltage sensorswhich could help to reduce the cost and complexity of the final system.

Another advantage of only requiring a single voltage sensor is that thisprovides redundancy. If the motor position can be entirely determinedusing neutral point voltage measurements, then the DC link voltagemeasurement could be used as a diagnostic cross check. If the controllerdetermines that the DC link voltage measurement does not agree with theneutral point voltage measurements in equations (16a)-(16c) sufficientlyto meet a consistency condition which is defined within the controller,then the controller can be arranged to generate an output indicatingthat there must be a fault in the system.

Sampling u_(n) in Three Inverter States

The main disadvantage with measuring six inverter states is that itgives the smallest available modulation index. This can be avoided bytaking measurements in only three non-zero inverter states, one fromeach phase, for example using the combination of states shown in FIG. 8.The inverter states may be all positive, all negative or a combinationof both. As with six inverter states, taking measurements in only threeinverter states makes it possible to determine motor position withoutmeasuring the DC link voltage. However, it increases the availablemodulation index for creating motor torque.

The neutral point voltage (for the +U inverter state) is defined byequation (19), which assuming zero speed/zero current conditions allowsthe three positive vector states +UVW to be defined by:

$\begin{matrix}{{{u_{n}\left( {+ U} \right)} \approx {\frac{U_{DC}}{3}\left\lbrack {1 - {\frac{\overset{}{L}}{\overset{\_}{L}}{\cos \left( {2\theta_{e}} \right)}}} \right\rbrack}}{{u_{n}\left( {+ V} \right)} \approx {\frac{U_{DC}}{3}\left\lbrack {1 - {\frac{\overset{}{L}}{\overset{\_}{L}}{\cos \left( {{2\theta_{e}} - \frac{4\pi}{3}} \right)}}} \right\rbrack}}{{u_{n}\left( {+ W} \right)} \approx {\frac{U_{DC}}{3}\left\lbrack {1 - {\frac{\overset{}{L}}{\overset{\_}{L}}{\cos \left( {{2\theta_{e}} - \frac{4\pi}{3}} \right)}}} \right\rbrack}}} & \left( {{23a},{23b},{23c}} \right)\end{matrix}$

Measuring the neutral point in three inverter states (one from each pairof complementary states) is advantageous as, for a motor with balancedphase resistances, there are no first order harmonics to cancel out(unlike the di/dt technique). Also, since it is a three phase system,the DC component and 3rd order harmonics are naturally removed. Thismeans that the three inverter state technique is not affected by changesin the back EMF, and therefore it is speed independent.

This allows the position to be calculated directly from the threeneutral point voltage measurements using:

$\begin{matrix}{{2\hat{\theta}} = {{{- \arg}\left\{ {u_{nU} + {u_{nV}^{j\frac{2\pi}{3}}} + {u_{nW}^{j\frac{4\pi}{3}}}} \right\}} + \pi}} & (24)\end{matrix}$

In reality there may be a requirement to improve the position byremoving the 1st and 4th order terms as, as the current increases andthe inductance saturates, the harmonic content of the motor will change.

An example of a control system using a sensorless position algorithmincorporating feedforward compensation as described above is shown inFIG. 9. A phase inductance module 100 is arranged to receive as inputsthe neutral point voltage measurements for the three active states, andthe neutral point voltage as measured or calculated. This module 100outputs the three phase inductances in the UVW frame, to which thecompensation terms are applied, in the UVW frame, by a compensationmodule 102. An inductance filter block 104 filters the inductance valuesand inputs them to a transformation block 106 which transforms theinductances to the alpha-beta frame. From the outputs of thetransformation block, A tan 2θ is calculated, from which the position θis calculated and then filtered by a position filter block 110. Theposition θ is fed back via a compensation filter block 112 to a harmoniccompensation calculation block 114 which also receives as an input the Qaxis current demand and calculates appropriate harmonic compensationterms for compensating the initially calculated phase inductances asdescribed above.

It will be appreciated that while this system is described in terms offunctional blocks which can correspond to separate processing steps, theprocessing which performs the functional steps may be combined togetherand can be performed in software or hardware.

As the transformation from UVW to alpha-beta frame does not require theposition the compensation could also be applied in the alpha-beta frame.Similarly while the position is shown as computed from arc-tangent; aPLL can be used just as easily.

FIG. 10 shows how the neutral point voltage, as measured during thethree active states described above with reference to FIG. 8, variesbetween different electrical positions of the motor. The left and rightsides of the figure correspond to two different positions. The bottomhalf of each side shows the phase terminal voltages during one PWMperiod, and the top half shows the neutral point voltage and how itvaries over the PWM period and between the three active states. It canbe seen that in this case in all three states the neutral point voltageis significantly different for the two positions, indicating that themethod of this embodiment can successfully be used to determine motorposition.

1. A control system for an AC electric motor which comprises a rotor anda stator and a plurality of phase windings connected in a starformation, each winding having one end connected to a common neutralpoint and another end arranged to have a terminal voltage applied to it,the control system comprising switching means arranged to control theterminal voltages applied to the windings and control means arranged tocontrol the switching means so as to switch it between a plurality ofstates in each of a sequence of PWM periods, wherein the control meansis further arranged to measure the voltage at the neutral point atsample times and to generate from the measured voltages an estimation ofthe rotational position of the rotor.
 2. A system according to claim 1wherein the control means is further arranged to measure a DC linkvoltage and to further use the DC link measurement to generate theposition estimation.
 3. A system according to claim 1 or claim 2 whereinthe control means is arranged to measure the neutral point voltage whenall of the phases are connected to the DC link voltage thereby by togenerate an estimation of the DC link voltage, and to use thatestimation to generate the position estimation.
 4. A system according toclaim 3 when dependent on claim 2 wherein the control means is arrangedto compare the measured DC link voltage with the estimated DC linkvoltage and if they fail to meet a consistency condition, to generate afault indication.
 5. A system according to any foregoing claim whereinthe sample times are within the PWM periods.
 6. A system according toany foregoing claim wherein the control means is arranged to measure theneutral point voltage during two complementary states within one PWMperiod.
 7. A system according to claim 6 wherein the control means isarranged to measure the neutral point voltage during two pairs of twocomplementary states within one PWM period.
 8. A system according to anyforegoing claim wherein the control means is arranged to measure theneutral point voltage during each of a number of non-complementarystates equal to the number of phases of the motor, within one PWMperiod.
 9. A system according to any foregoing claim wherein the controlmeans is arranged to define a minimum state time for neutral pointvoltage measurement and a minimum number of active states within a PWMperiod in which the neutral point voltage needs to be measured, todetermine a demanded net voltage for the PWM period, and to control theswitching means so as to provide at least the minimum number of activestates, each for at least the minimum state time, and to provide thedemanded net voltage.
 10. A method of determining the rotationalposition of an AC machine having a plurality of windings each connectedat one end to a neutral point and at the other end to switching means,the switching means being switchable between a plurality of states tocontrol voltages applied to the windings, the method comprisingmeasuring the voltage at the neutral-point in at least two states of theswitching means and estimating motor position from the measuredvoltages.
 11. A control system for an AC electric motor which comprisesa rotor and a stator and a plurality of phase windings connected in astar formation, each winding having one end connected to a commonneutral point and another end arranged to have a terminal voltageapplied to it, the control system comprising switching means arranged tocontrol the terminal voltages applied to the windings and control meansarranged to control the switching means so as to switch it between aplurality of states in each of a sequence of PWM periods, wherein thecontrol means is further arranged to measure the voltage at the neutralpoint when all of the phases are connected to the DC link voltagethereby by to generate an estimation of the DC link voltage.
 12. Acontrol system according to claim 11 wherein the control means isfurther arranged to compare a measured DC link voltage with theestimated DC link voltage and if they fail to meet a consistencycondition, to generate a fault indication.